Re: Function composition of natural transformations?

[Tim Porter <t.porter@bangor.ac.uk>]

Toby Bartels wrote:

>Steve Vickers wrote:
>
>
>
>>It is also possible to use a 2-dimensional syntax, in which horizontal
>>composition is composed horizontally and vertical composition is
>>composed vertically. Then algebraic manipulations are a bit like sliding
>>tiles around in a tray.
>>
>>
>
>Of course this can be done using big diagrams.
>But is there a tight syntax for this just using text?
>Can you point to an example? (preferably a TeX source online,
>but a printed page in a regular journal would also work).
>
>
>-- Toby
>
>
Dear All,

In reply to Toby Bartels, there are various models of higher categories
in which the syntax is well attested and the `sliding of tiles' is
algebraically described.  One possible one that extends to arbitrary
dimensions is given in the paper:


        AL-AGL, A.A., BROWN, R. & STEINER, R., Multiple categories: the
        equivalence of a globular and a cubical approach, Advances in
        Math. 170 (2002) 71-118.

The links between `cubical' syntax and a more globular syntax are at the
heart of the extensive work on the equivalence between the various
models for weak n-categories. One problem is that there are no normal
forms for elements. In fact I think (possibly!) that the problem of
rewriting in these higher dimensional settings needs a higher
dimensional rewriting systems, and to model that one needs n-cateories
(and so on!)

Tim Porter